Isospectral potentials and conformally equivalent isospectral metrics on spheres, balls and Lie groups

We construct pairs of conformally equivalent isospectral Riemannian metrics $\phi_1 g$ and $\phi_2 g$ on spheres $S^n$ and balls $B^{n+1}$ for certain dimensions $n$, the smallest of which is $n=7$, and on certain compact simple Lie groups. In the case of Lie groups, the metric $g$ is left-invariant. In the case of spheres and balls, the metric $g$ is not the standard metric but may be chosen arbitrarily close to the standard one. For the same manifolds $(M,g)$ we also show that the functions $\phi_1$ and $\phi_2$ are isospectral potentials for the Schr\"odinger operator $\hbar^2\Delta +\phi$. To our knowledge, these are the first examples of isospectral potentials and of isospectral conformally equivalent metrics on simply connected closed manifolds.Comment: 34 pages, AMS-TeX; revised subsection 5.

Spectral isolation of bi-invariant metrics on compact Lie groups

We show that a bi-invariant metric on a compact connected Lie group $G$ is spectrally isolated within the class of left-invariant metrics. In fact, we prove that given a bi-invariant metric $g_0$ on $G$ there is a positive integer $N$ such that, within a neighborhood of $g_0$ in the class of left-invariant metrics of at most the same volume, $g_0$ is uniquely determined by the first $N$ distinct non-zero eigenvalues of its Laplacian (ignoring multiplicities). In the case where $G$ is simple, $N$ can be chosen to be two.Comment: 10 pages, new title, revised abstract and introduction, minor typos corrected, to appear in Ann. Inst. Fourier (Grenoble

Isospectral deformations of closed Riemannian manifolds with different scalar curvature

We construct the first examples of continuous families of isospectral Riemannian metrics that are not locally isometric on closed manifolds, more precisely, on $S^n\times T^m$, where $T^m$ is a torus of dimension $m\ge 2$ and $S^n$ is a sphere of dimension $n\ge 4$. These metrics are not locally homogeneous; in particular, the scalar curvature of each metric is nonconstant. For some of the deformations, the maximum scalar curvature changes during the deformation.Comment: amstex, 10 pages, no figure

Spectral isolation of naturally reductive metrics on simple Lie groups

We show that within the class of left-invariant naturally reductive metrics $\mathcal{M}_{\operatorname{Nat}}(G)$ on a compact simple Lie group $G$, every metric is spectrally isolated. We also observe that any collection of isospectral compact symmetric spaces is finite; this follows from a somewhat stronger statement involving only a finite part of the spectrum.Comment: 19 pages, new title and abstract, revised introduction, new result demonstrating that any collection of isospectral compact symmetric spaces must be finite, to appear Math Z. (published online Dec. 2009

(Re)constructing Dimensions

Compactifying a higher-dimensional theory defined in R^{1,3+n} on an n-dimensional manifold {\cal M} results in a spectrum of four-dimensional (bosonic) fields with masses m^2_i = \lambda_i, where - \lambda_i are the eigenvalues of the Laplacian on the compact manifold. The question we address in this paper is the inverse: given the masses of the Kaluza-Klein fields in four dimensions, what can we say about the size and shape (i.e. the topology and the metric) of the compact manifold? We present some examples of isospectral manifolds (i.e., different manifolds which give rise to the same Kaluza-Klein mass spectrum). Some of these examples are Ricci-flat, complex and K\"{a}hler and so they are isospectral backgrounds for string theory. Utilizing results from finite spectral geometry, we also discuss the accuracy of reconstructing the properties of the compact manifold (e.g., its dimension, volume, and curvature etc) from measuring the masses of only a finite number of Kaluza-Klein modes.Comment: 23 pages, 3 figures, 2 references adde

Complex morphology and functional dynamics of vital murine intestinal mucosa revealed by autofluorescence 2-photon microscopy

The mucosa of the gastrointestinal tract is a dynamic tissue composed of numerous cell types with complex cellular functions. Study of the vital intestinal mucosa has been hampered by lack of suitable model systems. We here present a novel animal model that enables highly resolved three-dimensional imaging of the vital murine intestine in anaesthetized mice. Using intravital autofluorescence 2-photon (A2P) microscopy we studied the choreographed interactions of enterocytes, goblet cells, enteroendocrine cells and brush cells with other cellular constituents of the small intestinal mucosa over several hours at a subcellular resolution and in three dimensions. Vigorously moving lymphoid cells and their interaction with constituent parts of the lamina propria were examined and quantitatively analyzed. Nuclear and lectin staining permitted simultaneous characterization of autofluorescence and admitted dyes and yielded additional spectral information that is crucial to the interpretation of the complex intestinal mucosa. This novel intravital approach provides detailed insights into the physiology of the small intestine and especially opens a new window for investigating cellular dynamics under nearly physiological conditions

On impact and volcanism across the Cretaceous-Paleogene boundary

The cause of the end-Cretaceous mass extinction is vigorously debated, owing to the occurrence of a very large bolide impact and flood basalt volcanism near the boundary. Disentangling their relative importance is complicated by uncertainty regarding kill mechanisms and the relative timing of volcanogenic outgassing, impact, and extinction. We used carbon cycle modeling and paleotemperature records to constrain the timing of volcanogenic outgassing. We found support for major outgassing beginning and ending distinctly before the impact, with only the impact coinciding with mass extinction and biologically amplified carbon cycle change. Our models show that these extinction-related carbon cycle changes would have allowed the ocean to absorb massive amounts of carbon dioxide, thus limiting the global warming otherwise expected from postextinction volcanism

New age constraints for the Salamanca Formation and lower Río Chico Group in the western San Jorge Basin, Patagonia, Argentina: Implications for Cretaceous-Paleogene extinction recovery and land mammal age correlations

The Salamanca Formation of the San Jorge Basin (Patagonia, Argentina) preserves critical records of Southern Hemisphere Paleocene biotas, but its age remains poorly resolved, with estimates ranging from Late Cretaceous to middle Paleocene. We report a multi-disciplinary geochronologic study of the Salamanca Formation and overlying Rio Chico Group in the western part of the basin. New constraints include (1) an 40Ar/39Ar age determination of 67.31 ± 0.55 Ma from a basalt flow underlying the Salamanca Formation, (2) micropaleontological results indicating an early Danian age for the base of the Salamanca Formation, (3) laser ablation HR-MC-ICP-MS (high resolution-multi collector-inductively coupled plasma-mass spectrometry) U-Pb ages and a high-resolution TIMS (thermal ionization mass spectrometry) age of 61.984 ± 0.041(0.074)[0.100] Ma for zircons from volcanic ash beds in the Penas Coloradas Formation (Rio Chico Group), and (4) paleomagnetic results indicating that the Salamanca Formation in this area is entirely of normal polarity, with reversals occurring in the Rio Chico Group. Placing these new age constraints in the context of a sequence stratigraphic model for the basin, we correlate the Salamanca Formation in the study area to Chrons C29n and C28n, with the Banco Negro Inferior (BNI), a mature widespread fossiliferous paleosol unit at the top of the Salamanca Formation, corresponding to the top of Chron C28n. The diverse paleobotanical assemblages from this area are here assigned to C28n (64.67–63.49 Ma), ∼2–3 million years older than previously thought, adding to growing evidence for rapid Southern Hemisphere floral recovery after the Cretaceous-Paleogene extinction. Important Peligran and “Carodnia” zone vertebrate fossil assemblages from coastal BNI and Penas Coloradas exposures are likely older than previously thought and correlate to the early Torrejonian and early Tiffanian North American Land Mammal Ages, respectively.Centro de Investigaciones Geográfica